> > >For every finite or infinite sequence of digits of d there is a rational > > >number including it.

WM is WRONG, AGAIN, AS USUAL!!!

For every finite sequence, but NOT for every infinite sequence.

The decimal (or binary or any other base for that matter) sequence which is 0 for non-prime fractional digit positions and 1 for the prime fractional digit positions has no"rational number including it".

> > > That can and has been proven.

> > Where has that been proven?

> Proof: Take any digit. For every n: There are infinitely many digits > following upon d_n, such that there are infinitely many rational numbers > identical with d_1, ..., d_n. Since this holds for *all* infinitely many > natural indices, there is no chance to identify 1/n! by any sequence of only > finitely indexed digits.

Are those "d_1, d_2,...,d_n" suposed to be single digits or digit sequnences for rational numbers? They seem to change meaning back and forth.

And for any natural n, 1/n! will have a repeating decimal expansion thus can be entirely determined by finitely many digits.

> > What rational number q has the property that: q includes sqrt(2).> > This has been proven False for All rational numbers.

> q is not sqrt(2), but every digit that can be calculated by a sequence with > limit sqrt(2) belongs to a rational approximation.

> > > But it leads to the correct result: It is impossible to distinguish d by > > > its digits from all entries of arationals-complete list.

It is quite possible to do it by a finitely expressed rule, like was done in the Cantor argument, ans such a finitely expressed rule completes a valid proof.

> > You haven't even defined what this means.

> To define what it means that two numbers differ by digits???> Sorry, if you don't know then we should stop here. > ....

One may have some idea of what it would mean from someone not tied up in WM's wild weird world of WMytheology, but since WM has a habit of using standard terms in non-standard ways without making clear what his non-standard meanings are, we do not take any of his phraseologies for granted without explicit explanations.

> > I have argued that a Finite Formula does what is required, but you just > > disagree and don't give any support for you argument other than "but > > infinitely many follow". This is Not Sufficient to prove your claim.

> A finite formula does what is required. SUM 1/n! is uniquely defined. Every > approximation can be calculated, that means every digit can be calculated.> Formula ==> digits. But from the digits which all are rational numbers the > formula cannot be obtained.

From seeing enough of those digits someone only a little smarter that WM might well deduce a valid finite formula. And any valid finite formula will do.--